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Unformatted text preview: ossible to ﬁnd explicit solutions to these partial differential equations under the simplest boundary conditions. For example, the
general solution to the one-dimensional wave equation
= c2 2
for the vibrations of an inﬁnitely long string, is
u(x, t) = f (x + ct) + g (x − ct),
where f and g are arbitrary well-behaved functions of a single variable.
Slightly more complicated cases require the technique of “separation of variables” together with Fourier analysis, as we studied before. Separation of variables reduces these partial diﬀerential equations to linear ordinary diﬀerential
equations, often with variable coeﬃcients. For example, to ﬁnd the explicit solution to the heat equation in a circular room, we will see that it is necessary
to solve Bessel’s equation.
The most complicated cases cannot be solved by separation of variables,
and one must resort to numerical methods, together with suﬃcient theory to
understand the qualitative behaviour of the s...
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This document was uploaded on 01/12/2014.
- Winter '14